Suppose you want to accumulate $10,000 for a down payment for a house. You will deposit $400 at the beginning of every month in an account that credits interest monthly at the rate of 0.6% per month. How long will it take you to achieve your goal?

 Suppose you want to accumulate $10,000 for a down payment for a house. You will deposit $400 at the beginning of every month in an account that credits interest monthly at the rate of 0.6% per month. How long will it take you to achieve your goal? 

Answer

Here

FV = 10000 , Per moth payments = 400, interest monthly = 0.6% =0.006

Number of years =?

10,000 = 400(1.006)+ 100(1.0075)n−1 + ... + 100(1.0075)

10,000 = 400(1.006)^n + 400(1.006)^n-1+…..+400(1.006)

Here a = 400(1.006)^n , x =1/1.006, Sn= 10,000

10,000 = 400(1.006)^n [1-1/1.006^n]/1-1/1.006

Now

1/1.006^n = 1.006^-n & 1/1.006 = 1.006^-1

10,000 = 400 (1.006)^n [1-1.006^-n]/1-1.006^-1

10,000 = 400(1.006)^n -400]/1-1.006^-1

Here

1-1.006^-1 = 0.005964

10,000 =400(1.006)^n -400]/ 0.005964

10,000 x 0.005964 = 400(1.006)^n -400

59.64 +400 = 400(1.006)^n

Divided by 400 both sides

459.64/400 = 400(1.006)^n/400

1.149105 = 1.006^n

Take ln both side

Ln(1.149105) = n ln(1.006)

Ln(1.149105)/ ln(1.006) =n

n = 23.23 is almost equal to 24 months